Question

A milkman has \(250\) litres of milk containing \(5\%\) fat. How many litres of milk containing \(15\%\) fat should he add to his stock so that the fat content in the mixture would be more than \(7\%\) but less than \(10\%\)?

• A. More than \(62.5\) litres but less than \(250\) litres
• B. More than \(100\) litres but less than \(200\) litres
• C. More than \(62.5\) litres but less than \(200\) litres
• D. More than \(100\) litres but less than \(250\) litres

Hint:

In mixture problems: \[ \text{New Percentage} = \frac{\text{Total amount of pure substance}} {\text{Total mixture}} \times 100 \] Form inequalities directly when the percentage is required to lie within a range.

Answer 1

The Correct Option is A

Concept: Percentage of fat in the mixture is \[ \frac{\text{Total fat}}{\text{Total quantity}} \times 100 \] Let \(x\) litres of \(15\%\) fat milk be added.

Step 1: Calculate the total fat in the mixture. Fat in \(250\) litres of \(5\%\) milk: \[ 250\times\frac{5}{100} = 12.5 \text{ litres} \] Fat in \(x\) litres of \(15\%\) milk: \[ x\times\frac{15}{100} = 0.15x \] Total fat: \[ 12.5+0.15x \] Total quantity: \[ 250+x \]

Step 2: Use the condition that fat percentage is more than \(7\%\). \[ \frac{12.5+0.15x}{250+x} \gt 0.07 \] \[\begin{aligned} 12.5+0.15x &\gt 17.5+0.07x \\ 0.08x &\gt 5 \\ x &\gt 62.5 \end{aligned}\]

Step 3: Use the condition that fat percentage is less than \(10\%\). \[ \frac{12.5+0.15x}{250+x} \lt 0.10 \] \[\begin{aligned} 12.5+0.15x &\lt 25+0.10x \\ 0.05x &\lt 12.5 \\ x &\lt 250 \end{aligned}\]

Step 4: Combine the inequalities. \[ 62.5\lt x\lt 250 \] \[\begin{aligned} \boxed{ 62.5\lt x\lt 250 } \end{aligned}\] Thus the milkman should add more than \(62.5\) litres but less than \(250\) litres. Hence, option \(\mathbf{(A)}\) is correct.